Breaking of axisymmetry and onset of unsteadiness in the wake of a sphere

The primary and secondary instabilities of the sphere wake are investigated from the viewpoint of nonlinear dynamical systems theory. For the primary bifurcation, a theory of axisymmetry breaking by a regular bifurcation is g iven. The azimuthal spectral modes are shown to coincide with nonlinear mod es of the instability, which provides a good reason for using the azimuthal expansion as an optimal spectral method. Thorough numerical testing of the implemented spectral-spectral-element discretization allows corroboration of existing data concerning the primary and secondary thresholds and gives their error estimates. The ideal axisymmetry of the numerical method makes it possible to confirm the theoretical conclusion concerning the arbitrarin ess of selection of the symmetry plane that arises. Investigation of comput ed azimuthal modes yields a simple explanation of the origin of the so-call ed bifid wake and shows at each Reynolds number the coexistence of a simple wake and a bifid wake zone of the steady non-axisymmetric regime. At the o nset of the secondary instability, basic linear and nonlinear characteristi cs including the normalized Landau constant are given. The periodic regime is described as a limit cycle and the power of the time Fourier expansion i s illustrated by reproducing experimental r.m.s. fluctuation charts of the streamwise velocity with only the fundamental and second harmonic modes. Ea ch time-azimuthal mode is shown to behave like a propagating wave having a specific spatial signature. Their asymptotic, far-wake, phase velocities ar e the same but the waves keep a fingerprint of their passing through the ne ar-wake region. The non-dimensionalized asymptotic phase velocity is close to that of an infinite cylinder wake. A reduced-accuracy discretization is shown to allow qualitatively satisfactory unsteady simulations at extremely low cost.